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## G = S3×D4×C9order 432 = 24·33

### Direct product of C9, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×D4×C9
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — S3×C2×C18 — S3×D4×C9
 Lower central C3 — C6 — S3×D4×C9
 Upper central C1 — C18 — D4×C9

Generators and relations for S3×D4×C9
G = < a,b,c,d,e | a9=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 400 in 178 conjugacy classes, 75 normal (39 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, D4, D4, C23, C9, C9, C32, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C2×D4, C18, C18, C3×S3, C3×S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C3×C9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, S3×C6, S3×C6, C62, S3×D4, C6×D4, S3×C9, S3×C9, C3×C18, C3×C18, C2×C36, D4×C9, D4×C9, C22×C18, S3×C12, C3×D12, C3×C3⋊D4, D4×C32, S3×C2×C6, C9×Dic3, C3×C36, S3×C18, S3×C18, S3×C18, C6×C18, D4×C18, C3×S3×D4, S3×C36, C9×D12, C9×C3⋊D4, D4×C3×C9, S3×C2×C18, S3×D4×C9
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, C9, D6, C2×C6, C2×D4, C18, C3×S3, C3×D4, C22×S3, C22×C6, C2×C18, S3×C6, S3×D4, C6×D4, S3×C9, D4×C9, C22×C18, S3×C2×C6, S3×C18, D4×C18, C3×S3×D4, S3×C2×C18, S3×D4×C9

Smallest permutation representation of S3×D4×C9
On 72 points
Generators in S72
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 55)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 28)(17 29)(18 30)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 46)(27 47)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 64)(45 65)
(1 38 20 31)(2 39 21 32)(3 40 22 33)(4 41 23 34)(5 42 24 35)(6 43 25 36)(7 44 26 28)(8 45 27 29)(9 37 19 30)(10 56 67 49)(11 57 68 50)(12 58 69 51)(13 59 70 52)(14 60 71 53)(15 61 72 54)(16 62 64 46)(17 63 65 47)(18 55 66 48)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 28)(8 29)(9 30)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 55)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)

G:=sub<Sym(72)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69), (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,55)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,28)(17,29)(18,30)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,46)(27,47)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,64)(45,65), (1,38,20,31)(2,39,21,32)(3,40,22,33)(4,41,23,34)(5,42,24,35)(6,43,25,36)(7,44,26,28)(8,45,27,29)(9,37,19,30)(10,56,67,49)(11,57,68,50)(12,58,69,51)(13,59,70,52)(14,60,71,53)(15,61,72,54)(16,62,64,46)(17,63,65,47)(18,55,66,48), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,28)(8,29)(9,30)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,55)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)>;

G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69), (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,55)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,28)(17,29)(18,30)(19,48)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,46)(27,47)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,64)(45,65), (1,38,20,31)(2,39,21,32)(3,40,22,33)(4,41,23,34)(5,42,24,35)(6,43,25,36)(7,44,26,28)(8,45,27,29)(9,37,19,30)(10,56,67,49)(11,57,68,50)(12,58,69,51)(13,59,70,52)(14,60,71,53)(15,61,72,54)(16,62,64,46)(17,63,65,47)(18,55,66,48), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,28)(8,29)(9,30)(10,56)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,55)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,52,49),(47,53,50),(48,54,51),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,55),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,28),(17,29),(18,30),(19,48),(20,49),(21,50),(22,51),(23,52),(24,53),(25,54),(26,46),(27,47),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,64),(45,65)], [(1,38,20,31),(2,39,21,32),(3,40,22,33),(4,41,23,34),(5,42,24,35),(6,43,25,36),(7,44,26,28),(8,45,27,29),(9,37,19,30),(10,56,67,49),(11,57,68,50),(12,58,69,51),(13,59,70,52),(14,60,71,53),(15,61,72,54),(16,62,64,46),(17,63,65,47),(18,55,66,48)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,28),(8,29),(9,30),(10,56),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,63),(18,55),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)]])

135 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C ··· 6I 6J 6K 6L 6M 6N ··· 6S 6T 6U 6V 6W 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E 12F 12G 18A ··· 18F 18G ··· 18X 18Y ··· 18AJ 18AK ··· 18AV 18AW ··· 18BH 36A ··· 36F 36G ··· 36L 36M ··· 36R order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 6 6 ··· 6 6 6 6 6 6 ··· 6 6 6 6 6 9 ··· 9 9 ··· 9 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 2 2 3 3 6 6 1 1 2 2 2 2 6 1 1 2 ··· 2 3 3 3 3 4 ··· 4 6 6 6 6 1 ··· 1 2 ··· 2 2 2 4 4 4 6 6 1 ··· 1 2 ··· 2 3 ··· 3 4 ··· 4 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C9 C18 C18 C18 C18 C18 S3 D4 D6 D6 C3×S3 C3×D4 S3×C6 S3×C6 S3×C9 D4×C9 S3×C18 S3×C18 S3×D4 C3×S3×D4 S3×D4×C9 kernel S3×D4×C9 S3×C36 C9×D12 C9×C3⋊D4 D4×C3×C9 S3×C2×C18 C3×S3×D4 S3×C12 C3×D12 C3×C3⋊D4 D4×C32 S3×C2×C6 S3×D4 C4×S3 D12 C3⋊D4 C3×D4 C22×S3 D4×C9 S3×C9 C36 C2×C18 C3×D4 C3×S3 C12 C2×C6 D4 S3 C4 C22 C9 C3 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 6 6 6 12 6 12 1 2 1 2 2 4 2 4 6 12 6 12 1 2 6

Matrix representation of S3×D4×C9 in GL4(𝔽37) generated by

 33 0 0 0 0 33 0 0 0 0 10 0 0 0 0 10
,
 10 0 0 0 0 26 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 1 0 0 0 0 1 36 0 0 2 36
,
 1 0 0 0 0 1 0 0 0 0 36 0 0 0 35 1
G:=sub<GL(4,GF(37))| [33,0,0,0,0,33,0,0,0,0,10,0,0,0,0,10],[10,0,0,0,0,26,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,1,2,0,0,36,36],[1,0,0,0,0,1,0,0,0,0,36,35,0,0,0,1] >;

S3×D4×C9 in GAP, Magma, Sage, TeX

S_3\times D_4\times C_9
% in TeX

G:=Group("S3xD4xC9");
// GroupNames label

G:=SmallGroup(432,358);
// by ID

G=gap.SmallGroup(432,358);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,192,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^3=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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